Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63027 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5,10,12)(6,13,8,11)(7,16,18,9)(14,15,17,19)(20,21,25,28)(22,29,30,26)(23,24,27,33)(31,35,34,32);(4,6,10,8)(5,11,12,13)(7,14,18,17)(9,15,16,19)(22,30)(26,29)(31,34)(32,35);(4,6,10,8)(5,11,12,13)(7,14,18,17)(9,15,16,19)(20,22)(21,26)(23,31)(24,32)(25,30)(27,34)(28,29)(33,35);(4,7)(5,9)(6,14)(8,17)(10,18)(11,15)(12,16)(13,19);(4,8,10,6)(5,11,12,13)(7,17,18,14)(9,15,16,19)(20,23)(21,24)(22,31)(25,27)(26,32)(28,33)(29,35)(30,34);(4,9)(5,7)(6,15)(8,19)(10,16)(11,14)(12,18)(13,17)(20,24)(21,27)(22,32)(23,28)(25,33)(26,34)(29,31)(30,35)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56015);